2. \( \sqrt[3]{8 x^{12} y^{15}} \) 4. \( \sqrt{2 x^{3}} \times \sqrt{8 x^{5}} \) 6. \( \sqrt[6]{64 a^{6} b^{12} c^{18}} \) 8. \( \sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} \) (10.) \( \sqrt{2 m^{3}} \times \sqrt{50 m x^{2}} \) 12. \( \sqrt[3]{3 x} \times \sqrt[3]{72 x^{5}} \)

Simplify the expression by following steps: - step0: Solution: \(64^{\frac{1}{6}}\left(a^{6}\right)^{\frac{1}{6}}\left(b^{12}\right)^{\frac{1}{6}}\left(c^{18}\right)^{\frac{1}{6}}\) - step1: Multiply the exponents: \(64^{\frac{1}{6}}\left(a^{6}\right)^{\frac{1}{6}}\left(b^{12}\right)^{\frac{1}{6}}c^{18\times \frac{1}{6}}\) - step2: Multiply the exponents: \(64^{\frac{1}{6}}\left(a^{6}\right)^{\frac{1}{6}}b^{12\times \frac{1}{6}}c^{18\times \frac{1}{6}}\) - step3: Multiply the exponents: \(64^{\frac{1}{6}}a^{6\times \frac{1}{6}}b^{12\times \frac{1}{6}}c^{18\times \frac{1}{6}}\) - step4: Multiply the numbers: \(64^{\frac{1}{6}}a^{1}b^{12\times \frac{1}{6}}c^{18\times \frac{1}{6}}\) - step5: Calculate: \(64^{\frac{1}{6}}ab^{12\times \frac{1}{6}}c^{18\times \frac{1}{6}}\) - step6: Multiply the numbers: \(64^{\frac{1}{6}}ab^{2}c^{18\times \frac{1}{6}}\) - step7: Multiply the numbers: \(64^{\frac{1}{6}}ab^{2}c^{3}\) - step8: Evaluate the power: \(2ab^{2}c^{3}\) Calculate or simplify the expression \( (3 * x)^(1/3) * (72 * x^5)^(1/3) \). Simplify the expression by following steps: - step0: Solution: \(\left(3x\right)^{\frac{1}{3}}\left(72x^{5}\right)^{\frac{1}{3}}\) - step1: Rewrite the expression: \(3^{\frac{1}{3}}x^{\frac{1}{3}}\times 72^{\frac{1}{3}}x^{\frac{5}{3}}\) - step2: Multiply the numbers: \(3^{\frac{1}{3}}\times 72^{\frac{1}{3}}x^{\frac{1}{3}}\times x^{\frac{5}{3}}\) - step3: Multiply the terms: \(3^{\frac{1}{3}}\times 72^{\frac{1}{3}}x^{2}\) - step4: Simplify: \(\sqrt[3]{3}\times 2\sqrt[3]{9}\times x^{2}\) - step5: Expand the expression: \(6x^{2}\) Calculate or simplify the expression \( (8)^(1/3) * (x^(12))^(1/3) * (y^(15))^(1/3) \). Simplify the expression by following steps: - step0: Solution: \(8^{\frac{1}{3}}\left(x^{12}\right)^{\frac{1}{3}}\left(y^{15}\right)^{\frac{1}{3}}\) - step1: Multiply the exponents: \(8^{\frac{1}{3}}\left(x^{12}\right)^{\frac{1}{3}}y^{15\times \frac{1}{3}}\) - step2: Multiply the exponents: \(8^{\frac{1}{3}}x^{12\times \frac{1}{3}}y^{15\times \frac{1}{3}}\) - step3: Multiply the numbers: \(8^{\frac{1}{3}}x^{4}y^{15\times \frac{1}{3}}\) - step4: Multiply the numbers: \(8^{\frac{1}{3}}x^{4}y^{5}\) - step5: Evaluate the power: \(2x^{4}y^{5}\) Calculate or simplify the expression \( \sqrt(27 * x^3 * y^5 / (12 * x^7 * y^3)) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{\frac{27x^{3}y^{5}}{12x^{7}y^{3}}}\) - step1: Divide the terms: \(\sqrt{\frac{9y^{2}}{4x^{4}}}\) - step2: Use the properties of radicals: \(\frac{\sqrt{9y^{2}}}{\sqrt{4x^{4}}}\) - step3: Simplify the expression: \(\frac{3y}{2x^{2}}\) Calculate or simplify the expression \( \sqrt(2 * x^3) * \sqrt(8 * x^5) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{2x^{3}}\times \sqrt{8x^{5}}\) - step1: Simplify the root: \(x\sqrt{2x}\times \sqrt{8x^{5}}\) - step2: Simplify the root: \(x\sqrt{2x}\times 2x^{2}\sqrt{2x}\) - step3: Calculate: \(x\times 2x^{2}\times 2x\) - step4: Multiply the terms: \(4x\times x^{2}\times x\) - step5: Calculate: \(4x^{3}\times x\) - step6: Calculate: \(4x^{4}\) Calculate or simplify the expression \( \sqrt(2 * m^3) * \sqrt(50 * m * x^2) \). Simplify the expression by following steps: - step0: Solution: \(\sqrt{2m^{3}}\times \sqrt{50mx^{2}}\) - step1: Simplify the root: \(m\sqrt{2m}\times \sqrt{50mx^{2}}\) - step2: Simplify the root: \(m\sqrt{2m}\times 5x\sqrt{2m}\) - step3: Calculate: \(m\times 5x\times 2m\) - step4: Multiply the terms: \(10mxm\) - step5: Calculate: \(10m^{2}x\) Here are the simplified results for each expression: 2. For \( \sqrt[3]{8 x^{12} y^{15}} \): \[ \sqrt[3]{8} = 2, \quad \sqrt[3]{x^{12}} = x^{4}, \quad \sqrt[3]{y^{15}} = y^{5} \] Therefore, the result is: \[ 2x^{4}y^{5} \] 4. For \( \sqrt{2 x^{3}} \times \sqrt{8 x^{5}} \): \[ \sqrt{2} \times \sqrt{8} = \sqrt{16} = 4, \quad \sqrt{x^{3}} \times \sqrt{x^{5}} = \sqrt{x^{8}} = x^{4} \] Therefore, the result is: \[ 4x^{4} \] 6. For \( \sqrt[6]{64 a^{6} b^{12} c^{18}} \): \[ \sqrt[6]{64} = 2, \quad \sqrt[6]{a^{6}} = a, \quad \sqrt[6]{b^{12}} = b^{2}, \quad \sqrt[6]{c^{18}} = c^{3} \] Therefore, the result is: \[ 2ab^{2}c^{3} \] 8. For \( \sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} \): \[ \frac{27}{12} = \frac{9}{4}, \quad \sqrt{x^{3}/x^{7}} = \sqrt{x^{-4}} = \frac{1}{x^{2}}, \quad \sqrt{y^{5}/y^{3}} = \sqrt{y^{2}} = y \] Therefore, the result is: \[ \frac{3y}{2x^{2}} \] 10. For \( \sqrt{2 m^{3}} \times \sqrt{50 m x^{2}} \): \[ \sqrt{2} \times \sqrt{50} = \sqrt{100} = 10, \quad \sqrt{m^{3}} \times \sqrt{m} = \sqrt{m^{4}} = m^{2}, \quad \sqrt{x^{2}} = x \] Therefore, the result is: \[ 10m^{2}x \] 12. For \( \sqrt[3]{3 x} \times \sqrt[3]{72 x^{5}} \): \[ \sqrt[3]{3} \times \sqrt[3]{72} = \sqrt[3]{216} = 6, \quad \sqrt[3]{x} \times \sqrt[3]{x^{5}} = \sqrt[3]{x^{6}} = x^{2} \] Therefore, the result is: \[ 6x^{2} \] In summary, the simplified results are: 1. \( 2x^{4}y^{5} \) 2. \( 4x^{4} \) 3. \( 2ab^{2}c^{3} \) 4. \( \frac{3y}{2x^{2}} \) 5. \( 10m^{2}x \) 6. \( 6x^{2} \)